Definition of Dot Product
v → , w → ∈ R n v → ⋅ w → = < v → , w → > = v → T w → = v 1 w 1 + v 2 w 2 + . . . + v n w n Example: v → = ( 1 2 ) , u → = ( 3 4 ) , u → ⋅ v → = 1 ∗ 3 + 2 ∗ 4 = 11 \overrightarrow{v}, \overrightarrow{w} \in \mathbb{R}^n \\
\overrightarrow{v} \cdot \overrightarrow{w} = <\overrightarrow{v}, \overrightarrow{w}> = \overrightarrow{v}^T \overrightarrow{w} = v_1 w_1 + v_2 w_2 + ... + v_n w_n \\
\text{Example: } \overrightarrow{v} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \overrightarrow{u} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}, \overrightarrow{u} \cdot \overrightarrow{v} = 1 * 3 + 2 * 4 = 11 \\ v , w ∈ R n v ⋅ w =< v , w >= v T w = v 1 w 1 + v 2 w 2 + ... + v n w n Example: v = ( 1 2 ) , u = ( 3 4 ) , u ⋅ v = 1 ∗ 3 + 2 ∗ 4 = 11
Properties of Dot Products
v → ⋅ w → = w → ⋅ v → ( u → + v → ) ⋅ w → = u → ⋅ w → + v → ⋅ w → ( λ u → ) ⋅ v → = λ ( u → ⋅ v → ) v → ⋅ v → ≥ 0 , v → ⋅ v → = 0 ⟺ v → = 0 \overrightarrow{v} \cdot \overrightarrow{w} = \overrightarrow{w} \cdot \overrightarrow{v} \\
(\overrightarrow{u} + \overrightarrow{v}) \cdot \overrightarrow{w} = \overrightarrow{u} \cdot \overrightarrow{w} + \overrightarrow{v} \cdot \overrightarrow{w} \\
(\lambda \overrightarrow{u}) \cdot \overrightarrow{v} = \lambda (\overrightarrow{u} \cdot \overrightarrow{v}) \\
\overrightarrow{v} \cdot \overrightarrow{v} \geq 0, \overrightarrow{v} \cdot \overrightarrow{v} = 0 \iff \overrightarrow{v} = 0 v ⋅ w = w ⋅ v ( u + v ) ⋅ w = u ⋅ w + v ⋅ w ( λ u ) ⋅ v = λ ( u ⋅ v ) v ⋅ v ≥ 0 , v ⋅ v = 0 ⟺ v = 0 Dot Products as Lengths/Norms
∣ ∣ v → ∣ ∣ = v → ⋅ v → = v 1 2 + v 2 2 + . . . + v n 2 Examples: v → = ( 1 2 3 ) , ∣ ∣ v → ∣ ∣ = 1 2 + 2 2 + 3 2 = 14 u → = ( 1 1 1 ) , ∣ ∣ u → ∣ ∣ = 1 2 + 1 2 + 1 2 = 3 w → = ( 3 4 ) , ∣ ∣ w → ∣ ∣ = 3 2 + 4 2 = 25 = 5 || \overrightarrow{v} || = \sqrt{\overrightarrow{v} \cdot \overrightarrow{v}} = \sqrt{v_1^2 + v_2^2 + ... + v_n^2} \\
\text{Examples: } \\
\overrightarrow{v} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, ||\overrightarrow{v}|| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14} \\
\overrightarrow{u} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, ||\overrightarrow{u}|| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \\
\overrightarrow{w} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}, ||\overrightarrow{w}|| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \\ ∣∣ v ∣∣ = v ⋅ v = v 1 2 + v 2 2 + ... + v n 2 Examples: v = ⎝ ⎛ 1 2 3 ⎠ ⎞ , ∣∣ v ∣∣ = 1 2 + 2 2 + 3 2 = 14 u = ⎝ ⎛ 1 1 1 ⎠ ⎞ , ∣∣ u ∣∣ = 1 2 + 1 2 + 1 2 = 3 w = ( 3 4 ) , ∣∣ w ∣∣ = 3 2 + 4 2 = 25 = 5 Applications
v → ∈ R n , the normalized vector ‾ v ^ = 1 ∣ ∣ v → ∣ ∣ ⋅ v → ∣ ∣ v ^ ∣ ∣ = ∣ ∣ 1 ∣ ∣ v → ∣ ∣ ⋅ v → ∣ ∣ = 1 ∣ ∣ v → ∣ ∣ ⋅ ∣ ∣ v → ∣ ∣ = 1 Example: u → = ( 1 1 1 ) , u ^ = 1 3 ⋅ ( 1 1 1 ) Note that ∣ ∣ λ v → ∣ ∣ = ∣ λ ∣ ⋅ ∣ ∣ v → ∣ ∣ For v → , w → ∈ R n , the distance between v and w is ∣ ∣ v → − w → ∣ ∣ \overrightarrow{v} \in \mathbb{R}^n, \text{ the \underline{normalized vector} } \hat{v} = \frac{1}{||\overrightarrow{v}||} \cdot \overrightarrow{v} \\
||\hat{v}|| = || \frac{1}{||\overrightarrow{v}||} \cdot \overrightarrow{v} || = \frac{1}{||\overrightarrow{v}||} \cdot ||\overrightarrow{v}|| = 1 \\
\text{Example: } \overrightarrow{u} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \hat{u} = \frac{1}{\sqrt{3}} \cdot \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \\
\text{Note that } || \lambda \overrightarrow{v} || = | \lambda | \cdot || \overrightarrow{v} || \\
\text{For } \overrightarrow{v}, \overrightarrow{w} \in \mathbb{R}^n, \text{ the distance between v and w is } || \overrightarrow{v} - \overrightarrow{w} || v ∈ R n , the normalized vector v ^ = ∣∣ v ∣∣ 1 ⋅ v ∣∣ v ^ ∣∣ = ∣∣ ∣∣ v ∣∣ 1 ⋅ v ∣∣ = ∣∣ v ∣∣ 1 ⋅ ∣∣ v ∣∣ = 1 Example: u = ⎝ ⎛ 1 1 1 ⎠ ⎞ , u ^ = 3 1 ⋅ ⎝ ⎛ 1 1 1 ⎠ ⎞ Note that ∣∣ λ v ∣∣ = ∣ λ ∣ ⋅ ∣∣ v ∣∣ For v , w ∈ R n , the distance between v and w is ∣∣ v − w ∣∣ Orthogonal Vectors
v → , w → ∈ R n are orthogonal if v → ⋅ w → = 0 , often denoted as v → ⊥ w → Example: v → = ( 1 1 ) , w → = ( 1 − 1 ) , v → ⋅ w → = 1 ∗ 1 − 1 ∗ 1 = 0 ; v → ⊥ w → \overrightarrow{v}, \overrightarrow{w} \in \mathbb{R}^n \text{ are orthogonal if } \overrightarrow{v} \cdot \overrightarrow{w} = 0 \text{, often denoted as } \overrightarrow{v} \perp \overrightarrow{w} \\
\text{Example: }
\overrightarrow{v} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \overrightarrow{w} = \begin{pmatrix} 1 \\ -1 \end{pmatrix}, \overrightarrow{v} \cdot \overrightarrow{w} = 1 * 1 - 1 * 1 = 0; \overrightarrow{v} \perp \overrightarrow{w} v , w ∈ R n are orthogonal if v ⋅ w = 0 , often denoted as v ⊥ w Example: v = ( 1 1 ) , w = ( 1 − 1 ) , v ⋅ w = 1 ∗ 1 − 1 ∗ 1 = 0 ; v ⊥ w Important Inequalities and Equalities
Traingle Inequality: ∣ ∣ v → + w → ∣ ∣ ≤ ∣ ∣ v → ∣ ∣ + ∣ ∣ w → ∣ ∣ Pythagorean Theorem: If v → ⊥ w → , then ∣ ∣ v → + w → ∣ ∣ 2 = ∣ ∣ v → ∣ ∣ 2 + ∣ ∣ w → ∣ ∣ 2 Parallelogram Law: ∣ ∣ v → + w → ∣ ∣ 2 + ∣ ∣ v → − w → ∣ ∣ 2 = 2 ∣ ∣ v → ∣ ∣ 2 + 2 ∣ ∣ w → ∣ ∣ 2 Cauchy-Schwarz Inequality: ∣ v → ⋅ w → ∣ ≤ ∣ ∣ v → ∣ ∣ ⋅ ∣ ∣ w → ∣ ∣ \text{Traingle Inequality: } || \overrightarrow{v} + \overrightarrow{w} || \leq || \overrightarrow{v} || + || \overrightarrow{w} || \\
\text{Pythagorean Theorem: If } \overrightarrow{v} \perp \overrightarrow{w} \text{, then } || \overrightarrow{v} + \overrightarrow{w} ||^2 = || \overrightarrow{v} ||^2 + || \overrightarrow{w} ||^2 \\
\text{Parallelogram Law: } || \overrightarrow{v} + \overrightarrow{w} ||^2 + || \overrightarrow{v} - \overrightarrow{w} ||^2 = 2|| \overrightarrow{v} ||^2 + 2|| \overrightarrow{w} ||^2 \\
\text{Cauchy-Schwarz Inequality: } | \overrightarrow{v} \cdot \overrightarrow{w} | \leq || \overrightarrow{v} || \cdot || \overrightarrow{w} || Traingle Inequality: ∣∣ v + w ∣∣ ≤ ∣∣ v ∣∣ + ∣∣ w ∣∣ Pythagorean Theorem: If v ⊥ w , then ∣∣ v + w ∣ ∣ 2 = ∣∣ v ∣ ∣ 2 + ∣∣ w ∣ ∣ 2 Parallelogram Law: ∣∣ v + w ∣ ∣ 2 + ∣∣ v − w ∣ ∣ 2 = 2∣∣ v ∣ ∣ 2 + 2∣∣ w ∣ ∣ 2 Cauchy-Schwarz Inequality: ∣ v ⋅ w ∣ ≤ ∣∣ v ∣∣ ⋅ ∣∣ w ∣∣